On the Performance of Sphere Decoding of Block
Codes.
Mostafa El-Khamy, Haris Vikalo, Babak Hassibi and R. J. McEliece
Electrical Engineering Department California Institute of Technology
Pasadena, CA 91125, USA
E-mail: mostafa, hvikalo, hassibi, [email protected]
Abstract— The performance of sphere decoding of block codes over a variety of channels is investigated. We derive a tight bound on the performance of maximum likelihood decoding of linear codes onq-ary symmetric channels. We use this result to bound the performance ofq-ary hard decision sphere decoders. We also derive a tight bound on the performance of soft decision sphere decoders on the AWGN channel for BPSK and M-PSK modulated block codes. The performance of soft decision sphere decoding of arbitraryfinite lattices or block codes is also analyzed.
I. INTRODUCTION ANDSUMMARY
A maximum likelihood decoder returns the closest code-word or lattice point to the received vector. Fincke and Pohst [1] developed a sphere decoder to solve the closest-point problem in general lattices.[2]. A faster sphere decoding algorithm was given by Schnorr and Euchner [3]. Algorithms based on sphere decoders are currently the state of the art for decoding and detection in multiple input multiple output linear channels [4], [5], [6]. One can also think of the Guruswami-Sudan (GS) decoder [7] as a sphere decoder for Reed Solomon (RS) codes whose radius can be larger than half the minimum distance. The radius of the sphere decoder provides a tradeoff between performance and complexity.
The performance of sphere decoding of linear block codes on additive white Gaussian noise channels (AWGN) and bi-nary symmetric channels (BSC) was analyzed in our previous work [8]. In this paper, we analyze the performance of linear block codes, defined over Fq, when transmitted over q-ary
symmetric channels (QSC) and the decoder is either the maximum likelihood decoder or a sphere decoder with an arbitrary search radius. This is done in Sec. II. These results are used to analyze the performance of RS codes on q-ary symmetric channels. In Sec. III, we derive tight bounds on the performance of sphere decoding of linear block codes with a binary or M-ary PSK modulation over an AWGN channel. We then show, in Sec. IV, how the performance of sphere decoding of an arbitrary code with an arbitrary modulation scheme (finite lattice) on AWGN channels can be analyzed. We illustrate the tightness of our analytic bounds by comparing them to numerical simulations.
II. SPHEREDECODING OFCODES INFq OVERq-ARY
SYMMETRICCHANNELS.
Consider an (n, k, d)linear code C over thefinitefield of q elements, Fq, transmitted over a q-ary symmetric channel
(QSC) and a sphere decoder which can correct τ symbol errors, where the symbols are inFq. For the case of RS codes,
the GS algorithm can correct up to τGS = n−
√ nk−1 symbol errors which is at least as big as the radius of the conventional Berlekamp-Massey algorithm, τBM = n−2k.
The bounded distance decoder error probability of RS codes has been previously studied (e.g. [9]).
Letsandp= (1−s)/(q−1)denote the success and error crossover probabilities of the QSC respectively. Transmitting a q-ary code over an AWGN channel followed by hard-decision can be modeled as transmitting it over an QSC. Assume that q = 2m, the channel alphabet size is 2b, b ≤ m, and each
q-ary symbol is mapped to m/b channel symbols. Letpc be
the probability that a channel symbol is incorrectly decoded, thens= (1−pc)m/b.
A. Bound on the ML decoding of linear block codes onq-ary
symmetric channels.
Let ζ be the Hamming distance between the transmitted codeword and the received word inFn
q. Throughout this paper
EM L will denote the event of an ML error. Then, similar to
the binary case [10], the ML error probability can be upper bounded as follows,
P(EM L)≤min
m {P(EM L, ζ < m) +P(ζ ≥m)}. (1)
Assuming that the code is linear, the probability that the receivedq-ary word lies outside a Hamming sphere (ball) of radiusm−1centered around the transmitted word is
P(ζ≥m) = n α=m n α (1−s)αsn−α. (2) The above equation will also provide a lower bound on the performance of the sphere decoder. The first term in (1) is upper bounded in the following lemma.
Lemma 1: For an (n, k, d) linear code over Fq, with a
channel with parameterssandp, P(EM L, ζ < m)≤ min{n,2(m−1)} w=d Gw min{w,m−1} α=0 w−α η=w−α 2 w! η!α!(w−η−α)!p η(1−p−s)αsw−η−α m−1−η−α β=0 n−w β (1−s)βsn−w−β ⎞ ⎠. (3)
Proof: We will assume that the all-zero codeword is transmitted. Now consider a codeword c with Hamming weight w and assume the received word r has a Hamming weight m−1. Consider the w nonzero symbols in c and the corresponding coordinates in r. Let r and c have the same symbols in η of these coordinates. Let α of these w coordinates in r be neither zero nor match those in c, and w −η − α of the remaining coordinates be zero. Since the Hamming weight of r is m−1, there must be m −1−η−α non-zero symbols in the remaining n−w coordinates and the remaining symbols will be zero. The probability of receiving such a word is w!
η!α!(w−η−α)!pη(1−p−
s)αsw−η−α n−w m−1−η−α
(1−s)m−1−η−αsn−w−(m−1−η−α).
In such a case, the Hamming distance between r and c is w+m −1−2η−α. An ML error results if this is less than the weight of r, i.e., if η ≥ w−α
2 . By summing
over all possible combinations of η and α and applying the union bound for all codewords that can be within a Hamming distance m from r, the error probability is upper bounded by 2(m−1) w=d Gw m−1 α=0 w−α η=w−α 2 w! η!α!(w−η−α)!pη(1−p−s)αsw−η−α n−w m−1−η−α (1 − s)m−1−η−α sn−w−(m−1−η−α). Applying the union bound
for all received words with Hamming weights less than m, m ≤m, the result follows.
One can now prove the following theorem,
Theorem 2: The ML error probability of an (n, k, d)q-ary linear code on a q-ary symmetric channel is upper bounded by P(EM L)≤ min{n,2(mo−1)} w=d Gw min{w,mo−1} α=0 w−α η=w−2α w! η!α!(w−η−α)!p η(1−p−s)αsw−η−α mo−1−η−α β=0 n−w β (1−s)βsn−w−β + n α=mo n α (1−s)αsn−α, wheremois the smallest integermsuch that
min{n,2m} w=d Gw min{w,m} α=0 q−2 q−1 α w−α η=w−α 2 1 q−1 η w! η!α!(w−η−α)! n−w m−η−α ≥ n m . (4)
It is worth noting that the optimum radius mo which
minimizes the bound on the ML error probability only depends on the weight enumerator of the code and the size of itsfinite field. Since the optimum radius does not depend on the SNR, it is valid forq-ary symmetric channels at any SNR. We also establish below a connection between mo and the covering
radius of the code.
Lemma 3: The covering radius of a linear code on Fq is
lower bounded bymo−1, wheremois given by Th. 2. Proof: DefineL(m)to be the left hand side term in (4) and co to be the all zero codeword. Similar to the proof of
Lem. 1, one can show that (q −1)mL(m) = |{r ∈ Fn q :
d(r,co) = m & d(r,ci)≤ m for some ci ∈ C \co}|.
Also, (q −1)mn m = |{r ∈ Fn q : d(r,co) = m}|. Since (q −1)mo−1L(mo−1) < (q −1)mo−1 n mo−1 , then there exit words r∈Fn
q such that minc∈Cd(r,c) = mo−1 and
this minimum is achieved when c is the all zero codeword co. By recalling that the covering radius is [11] Rc =
maxr∈Fn
q minc∈Cd(r,c),it follows thatRc≥mo−1.
Corollary 4: For any linear(n, k)code mo≤n−k+ 1. B. Sphere decoding of linear block codes onq-ary symmetric
channels.
Let HSD(m−1) denote a (hard decision) sphere decoder with radiusm−1that correctly decodes the received word if its Hamming distance from the transmitted word is less than m. Letd(y,v)be the Hamming distance between y andv, then ify∈Fn
q is received, the output from the decoder is
ˆ
c= arg min
v∈C d(y,v) (5)
subject to d(y,v)< m.
Using Gallager’s bounding technique [12], the error plus failure probability of the sphere decoder,P(Em), can be upper-bounded as follows
P(Em) = P(Em, ζ < r) +P(Em, ζ≥r)
≤ min
r<m{P(EM L, ζ < r) +P(ζ≥r)}, (6)
which follows from the fact that the sphere decoder performs ML decoding within the specified search radius.
Theorem 5: The performance of HSD(m−1) decoding of an(n, k, d)linear code, with a weight spectrum Gw, over a
q-ary symmetric channel, with a success probabilitys and a crossover probabilityp= (1−s)/(q−1), is upper bounded by
P(Em)≤
P(EM L, ζ < mo) +P(ζ ≥mo), m≥mo;
P(EM L, ζ < m) +P(ζ≥m), m < mo,
where mo is radius that minimizes (1) and is given by Th.
2. P(ζ ≥ m)is given by (2) and P(EM L, ζ < m)is upper bounded by (3).
C. Numerical Examples
In Fig. 1, the binary image of the(15,3)RS code is BPSK modulated over an AWGN channel. For16-ary hard decisions, the channel is modelled as an QSC. The performance bound
1 2 3 4 5 6 7 8 9 10−4 10−3 10−2 10−1 100 SNR (dB)
Codeword Error Rate
Performance of 16−ary HSD of (15,3) RS codes, BPSK on AWGN H−ML, bnd E(9), bnd F(9), bnd H−ML, sim E(9), sim F(9), sim GS ,bnd GS, sim S−ML, bnd
Fig. 1. The(15,3)RS code is BPSK modulated and transmitted over an AWGN channel. For the16-ary hard-decision decoder, the channel is an QSC.
of the hard ML (H-ML) decoder is shown ( Th. 2) and is the same as an HSD of radius 9. The bounds of (2) and (3) are also shown and labeled as F(9) and E(9) respectively. As seen, the three bounds (‘bnd’) are in close agreement with the simulation (‘sim’), for such a hypothetical sphere decoder. The error probability of the GS decoder with radius8is simulated and agrees with the bounded of Th. 5. For reference proposes, we show the average error probability of the soft decision bit level ML (S-ML) decoder (this is analyzed in [13]) which has about4dB gain over the symbol H-ML decoder.
III. SPHEREDECODINGBOUNDS FORPSK BLOCK
CODEDMODULATION
Consider a sphere decoder when the modulation is M-ary or binary phase shift keying (PSK) [14] and each transmitted codeword in the code has the same energy when mapped to the PSK constellation. Complex sphere decoding algorithms which solve the closest point search problem were developed in [15]. We will derive a bound on the performance of the corresponding soft decision sphere decoder for BPSK modulation which is tighter than our previous bound [8]. We show how this bound is applied for the case of M-ary PSK modulation. We will assume that the modulated code is linear. Note that the original code need not be binary. For example, an RS code defined overF2m could be mapped directly to an
2m-ary PSK constellation by a one-to-one mapping from the
symbols inF2m to the2m points in the PSK constellation.
We will introduce some notation, so the bound derived here is readily applicable for both BPSK and M-PSK modulation. Each codeword of lengthnwill be mapped to a word ofM -PSK symbols. If the code is binary, then eachlog2(M)bits are mapped to anM-ary symbol. The number of channel symbols will be denoted bync; for a binary code of lengthnand
M-PSK modulation,nc=n/log2(M)(For BPSK,nc=n.) Let
Gw be the number of codewords which are at an Euclidian
distanceδw from each other. For QPSK modulation and Gray
encoding [14],δw = √
2w, wherew is the (binary) Hamming distance between the codewords. For BPSK,δw= 2
√ w. Let nd denote the dimension of the considered space. For BPSK
Fig. 2. The coneVφintersects the sphereΩD,D >√ncsin(φ).
and M-PSK, nd = nc andnd = 2nc respectively. The code
will have the property that all codewords are of equal energy and lie on a sphere of radius√nc from the origin of space.
Consider a soft decision sphere decoder with an Euclidean decoding radius D, SSD(D). ΩD will denote annd
dimen-sional sphere centered around the transmitted codeword (all zero codeword) whileVθ will denote annddimensional right
circular cone with half angleθ. Following Gallager’s bounding technique and defining the regionΛ(θ, D)∆={Vθ∩ΩD} the
error plus failure probability of SSD(D) is upper bounded by P(ED)≤ min
θ {P(ED|z∈Λ(θ, D))P(z∈Λ(θ, D))
+P(z∈/Λ(θ, D))}, (7) wherez is thend dimensional noise of varianceσ2.
The ML error probability for the case of BPSK and M-ary modulation is tightly upper bounded by the Poltyrev tangential sphere bound by [10], [16]
P(EM L)≤P(EM L,z∈Vφ) +P(z∈/Vφ),
where tan(φ) = rφ/nc. By defining θb(ro) =∆
cos−1 δb/2 √ ro(1−δ2b/4nc)
,rφis the solution forroin this
equa-tion [16] b>0Gb(ro) θb(ro) 0 sinnd−3(ϑ)dϑ = √ πΓ(nd2−2) Γ(nd−1 2 ) .
Gb(ro)is equal toGb ifδ2b/4< ro(1−δ2b/4nc)and is zero
otherwise. From (7), one can prove the following theorem.
Theorem 6: The performance of SSD(D) for BPSK or MPSK modulation is upper bounded by
P(ED)≤ ⎧ ⎨ ⎩ P(EM L,z∈ΩD) +P(z∈/ΩD), D≤ √ncsin(φ); P(EM L,z∈Λ(φ, D)) +P(z∈/ΩD)+ P({z∈/Vφ} ∩ {z∈ΩD}), D >√ncsin(φ) . We will call Dφ=√ncsin(φ)the critical decoding radius.
We will now give expressions for the different terms that appeared in the theorem;
where the regularized Gamma functionΓr is given in terms
of the Gamma functionΓby Γr(v/2, w/2) = w 0 t v/2−1e−t/2 2v/2Γ(v/2)dt, w≥0; 0, w <0. . (9)
The joint probability of an ML error andz∈ΩD is
P(EM L,z∈ΩD) = (10) b: 0<δb/2<DGb √D bN(zo)Γr n d−1 2 ,D 2− z2o 2σ2 dzo, whereN(z) = √1 2πσ2e−z 2/2σ2
is the normal distribution. Define ya(φ) and yb(φ) to be the altitudes at which the
cone Vφ intersects the sphere ΩD (see Fig. 2).. It follows
that ya,b(φ) = √nc(1−2Ua,b(φ, D)), where Ua,b(θ, D) = 4nc±√16nc2−16ncsec2(θ)(nc−D2)
8ncsec2(θ) . The cone Vφ intersects ΩD
ifD >√ncsin(φ)at which P({z∈/Vφ} ∩ {z∈ΩD}) = yb(φ) ya(φ)N(z1) Γr nd−1 2 , ω2z1 2σ2 − Γr nd−1 2 , rz21(φ) 2σ2 N(z1)dz1, whereω2z1 =D2−z12andrz1(φ) ∆ =√rφ 1−√zn1c . Consider a codeword at a distanceδw, then the half angle of the cone
bisecting this distance is θw = sin−1(δw/2√nc). This cone
will intersect the sphere ΩD at altitudes xa(w) and xb(w)
given by xa,b(w) = √nc(1−2Ua,b(θw, D)). Let βz1(w)
∆ = √ nc−z1 r 4nc δ2w−1
. Now define the integrals I(γ, w, z1) ∆= N(z1)βγ z1(w)N(z2)Γr nd−2 2 ,γ 2−z2 2 2σ2 dz2, and I2(w) =xyaa((wφ))I(ωz1, w, z1)dz1+ yb(φ) ya(φ)I(rz1(φ), w, z1)dz1+ xb(w) yb(φ) I(ωz1, w, z1)dz1.
Taking the union over all the non-zero Euclidean weights, it follows that forD >√ncsin(φ),
P(EM L,z∈Λ(φ, D)) =
w>0
Gw(rφ)I2(w). (11)
It is to be noted that the same equations hold whether D > √
ncor √ncsin(φ)< D≤ √nc.
In Fig. 3, we show how the bounds derived for M-ary modulated spherical codes are tight. A codeword in the (24,12) Golay code is mapped into 12 QPSK symbols and transmitted over AWGN channel. As observed, the simulated performance of the ML decoder and the SD sphere decoder are tightly bounded by the bounds given in this section. The critical decoding radius in the 2×12 dimensional space is Dφ= 2.667.
IV. SPHEREDECODING OFFINITELATTICES
In this section, we consider the case of soft decision sphere decoding of a general finite lattice or code C. The code is not constrained to be a linear code and the transmitted codewords are not constrained to have a fixed energy The channel symbols of a transmitted codeword are also not required to have the same energy. Define Gw(i) to be the
number of mapped codewords with an Euclidean distanceδw
from the ith codeword. Given that ci is transmitted, let the
error probability of SSD(D) be upper bounded byPi(ED). By
taking the expectation over all codewords, P(ED)≤ 1
|C|
ci∈C
Pi(ED). (12)
Now, if we assume that Pi(ED) is of the union bound
form; Pi(ED) =
w>0Gw(i)Pi(w)(ED), wherePi(w)(ED)is
the probability of a sphere decoder error due to incorrectly decoding a codeword at a distanceδw whenciis transmitted.
The error probability of SSD(D) can thus be upper bounded by P(ED) ≤ w:δ
w>0G¯wP
(w)(E
D), where P(w)(ED) is
the probability that the sphere decoder erroneously decodes a codeword at a distance w from the transmitted codeword and ¯ Gw= 1 |C| ci∈C Gw(i), (13)
is the average number of codewords which are at an Euclidean distanceδw from another codeword. The ML error probability
was analyzed for such a case by Hughes [17], [18]. The optimum radius that minimizes the Hughes upper bound on the ML error probability will be denoted byDo.
Theorem 7: The error (plus failure) probability of SSD(D) of an arbitraryfinite lattice or code is upper bounded by P(ED)≤
P(EM L,z∈ΩD) +P(z∈/ΩD), D < Do;
P(EM L,z∈ΩDo) +P(z∈/ ΩDo), D≥Do.
, where Dois the root of the equation
w: 0<δw2 <D ¯ Gw θw,D 0 sin(θ) nd−2dθ= √πΓ nd−1 2 Γnd 2 , (14) andθw,d= cos−1(δw/2D).
The theorem follows by observing that SSD(D) is equiva-lent to the maximum likelihood decoder if the received word falls within an Euclidean distanceDfrom the transmitted one. The bound developed here is universal in the sense that also applies for the case of a linear code with equal energy codewords. However, it is to be noted that the Hughes bound on ML decoding is not tighter than the Poltyrev tangential sphere bound [19] which implies that for MPSK and BPSK modulated schemes the bound of Th. 6 is tighter than that of Th. 7.
For the case of M-PSK modulation of a linear code, the constellation may not result in a Hamming space if M >4. In such a case the ensemble average weight enumerator G¯w (13) can be used with the bounds of Sec. III to analyze the performance. The same technique can be used with the results in Sec. II for general nonlinear lattices transmitted overq-ary symmetric channels.
A. Numerical Example
Assume an (15,3) RS code over F16 and assume a
one-to-one mapping from the symbols ofF16to the points of an
16-QAM modulation [14], whose average energy per symbol is10. The resulting lattice is no longer linear, meaning that it
2 4 6 8 10 12 10−12 10−10 10−8 10−6 10−4 10−2
100 Sphere Decoding of (24,12) Golay Code, QPSK modulation over AWGN
SNR (dB)
Codeword Error Rate
SSD(2), sim SSD(Dφ), sim SSD(3), sim ML, sim SSD(2), bnd SSD(Dφ), bnd SSD(3), bnd ML, bnd
Fig. 3. Bounds on the performance of soft-decision sphere decoding of the
(24,12)Golay code when QPSK modulated over an AWGN channel.
0 2 4 6 8 10−4 10−3 10−2 10−1 100 SNR
Codeword Error Rate
P(EML, z ∈ ΩD), sim P(ED), sim P(EML, z ∈ ΩD), bnd P(ED), bnd 1 2 3 4 5 6 10−3 10−2 10−1 100 SNR
Codeword Error Rate
SSD Bounds for 16−QAM modulated (15,3) RS codes
P(EML, z ∈ ΩD), sim
P(ED), sim P(EML, z ∈ ΩD), bnd
P(ED), bnd
Fig. 4. The(15,3)RS code is 16-QAM modulated and transmitted over an AWGN channel. The sphere decoder is a soft decision sphere decoder with an Euclidean radius10(left) andDo = 12.9(right). The bounds are compared to simulations for a sphere decoding ML error and the error plus failure probability.
is not necessary thatGw(ci) =Gw(cj)ifi=j. Furthermore,
the codewords (lattice points) are not of equal energy. The ensemble weight enumerator G¯w was numerically computed to evaluate the bounds. The radius that minimizes the bound on the ML error probability is Do = 12.9. In Fig. 4, we
confirm that the bounds on the sphere decoder error probability agree with the simulations for the cases of D = 10 and D = Do. We also compare the simulated performance of
ML error probabilityP(EM L,z∈ΩD)to that of the analytic performance in both cases. At low SNRs this probability is low as the probability of the received word falling inside the sphere is relatively low. As more received words fall inside the sphere, the ML error probability increases as the SNR increases. At a certain SNR, the probability of the ML error starts decreasing due to the improved reliability of the received word.
V. CONCLUSIONS
Bounds on the error plus failure probability of hard-decision and soft-decision sphere decoding of block codes were
de-rived. By comparing with the simulations of the corresponding decoders these bounds are tight. The ML performance of codes on q-ary symmetric channels is analyzed. The performance of sphere decoding of Reed Solomon codes and their binary images was analyzed. Moreover, the bounds are extremely useful in predicting the performance of the sphere decoders at the tail of error probability when simulations are prohibitive. The bounds allows one to pick the radius of the sphere decoder that fits best the performance, throughput and complexity requirements of the system.
ACKNOWLEDGMENT
This research was supported by NSF grant no. CCF-0514881 and grants from Sony, Qualcomm, and the Lee Center for Advanced Networking.
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